Copyright © Cengage Learning. All rights reserved. 4 Integration Copyright © Cengage Learning. All rights reserved.

The Natural Logarithmic Function: Integration 4.6 Copyright © Cengage Learning. All rights reserved.

Objectives Use the Log Rule for Integration to integrate a rational function. Integrate trigonometric functions.

Log Rule for Integration

Log Rule for Integration We have studied two differentiation rules for logarithms. The differentiation rule d/dx [ln x] = 1/x produces the Log Rule for Integration. The differentiation rule d/dx [ln u] = u /u produces the integration rule  1/u du = ln| u | + C. These rules are summarized below.

Log Rule for Integration Because du = u dx, the second formula can also be written as Alternative form of Log Rule

Example 1 – Using the Log Rule for Integration To find  1/(2x) dx, let u = 2x. Then du = 2 dx. Multiply and divide by 2. Substitute: u = 2x. Apply Log Rule. Back-substitute.

Log Rule for Integration With antiderivatives involving logarithms, it is easy to obtain forms that look quite different but are still equivalent. For instance, both ln| (3x + 2)1/3 | + C and ln| 3x + 2 |1/3 + C are equivalent to the antiderivative. Integrals to which the Log Rule can be applied often appear in disguised form. For instance, when a rational function has a numerator of degree greater than or equal to that of the denominator, division may reveal a form to which you can apply the Log Rule.

Log Rule for Integration To master integration technique, you must recognize the “form-fitting” nature of integration. In this sense, integration is not nearly as straightforward as differentiation. Differentiation takes the form “Here is the question; what is the answer?” Integration is more like “Here is the answer; what is the question?”

Log Rule for Integration The following are the guidelines you can use for integration.

Integrals of Trigonometric Functions

Integrals of Trigonometric Functions We have looked at six trigonometric integration rules—the six that correspond directly to differentiation rules. With the Log Rule, you can now complete the set of basic trigonometric integration formulas.

Example 8 – Using a Trigonometric Identity Find Solution: This integral does not seem to fit any formulas on our basic list. However, by using a trigonometric identity, you obtain

Example 8 – Solution cont’d Knowing that Dx [cos x] = – sin x, you can let u = cos x and write Apply trigonometric identity and multiply and divide by –1. Substitute: u = cos x. Apply Log Rule. Back-substitute.

Integrals of Trigonometric Functions Example 8 uses a trigonometric identity to derive an integration rule for the tangent function. The integrals of the six trigonometric functions are summarized below.